This paper proposes anytime feasible networked dynamics to solve resource allocation problems over time-varying multi-agent networks. The state of agents represents the assigned resources while their total (equal to demand) is constant. The idea is to optimally allocate the resources among the group of agents by minimizing the overall cost subject to fixed sum of resources. Each agent's information is local and restricted to its own state, cost function, and the ones from its immediate neighbors. This work provides a fast convergent solution (compared to linear dynamics) while considering more-relaxed uniform network connectivity and (logarithmic) quantized communications among agents. The proposed dynamics reaches optimal solution over switching (sparsely-connected) undirected networks as far as their union over some bounded non-overlapping time-intervals has a spanning tree. Moreover, we prove anytime-feasibility of the solution, uniqueness, and convergence to the optimal value irrespective of the specific nonlinearity in the proposed dynamics. Such general proof analysis applies to many similar 1st-order allocation dynamics subject to strongly sign-preserving nonlinearities, e.g., actuator saturation in generator coordination. Further, anytime feasibility (despite the nonlinearities) ensures that our solution satisfies the fixed-sum resources constraint at all times.